As previously mentioned, for Schwarz preconditioning to be effective, it is essential that every combination of basis functions that can become almost linearly dependent is contained in an index block, which is generally achieved by selecting these blocks based on the overlap in the supports of basis functions. In [1]}, an index block is devised for every cut element, containing all basis functions supported on it. For basis functions that are not supported on any cut element, a simple diagonal scaling is performed.
A tailored block selection procedure for multilevel \(hp\) -adaptive discretizations is developed in [2]}. In this procedure, the set of basis functions supported on a refined (leaf) element is further restricted to only the necessary DOFs. Additionally, the procedure is optimized by only devising blocks for cut elements with a volume fraction that is smaller than a certain threshold \(\eta ^* \in [0,1]\) . For locally refined discretizations based on truncated hierarchical B-splines, a block selection strategy is presented in [3]}.
In [1]} it is noted that, for vector-valued problems, degrees of freedom describing the solution in a certain geometrical dimension can generally not form a linear dependency with degrees of freedom describing the solution in another geometrical dimension. Hence, it can be beneficial to generate separate index blocks for each geometrical dimension.
